SwiftAccelerate

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SwiftAccelerate

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A playground demoing how to use Accelerate & Swift for Linear Algebra (vector/matrix manipulations)

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README.md

// repository documentation

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SwiftAccelerate
===============

This note provides a concise tutorial on how you may use Apple's `Accelerate` framework with the Swift programming language to perform vector/matrix manipulations, including matrix transposes, dot products, matrix inversions, etc. A playground illustrating all the functions discussed here is included. Also, a better formatted version can be accessed at [here](https://swift.versify-app.com/post/usehnl/).

Linking Your Project Against `Accelerate`
-----------------------------------

* Select your project by clicking on the blue icon in the top left corner. 
* In the `TARGETS` list (the panel in the middle), select the target you're compiling and then activate the `Build Phases` tab. 
* Click on the little triangle in front of `Link Binary With Libraries`.
* Click on the `+` sign and select ``Accelerate.framework`` in the popup.

Importing `Accelerate`
--------------------

Now that your project is linked against `Accelerate`, you can import it in your `.swift` file by issuing:

import Accelerate

Vector & Scalars
--------------

The general syntax for adding a scalar to a vector and for multiplying or dividing a vector by a scalar is as follows

vDSP_vs***D(vector, 1, &scalar, &result, 1, length_of_vector)

The `1`s tells the function to operate on each element of the vector. If you replace `1` with `2`, it'll operate on every other element instead. Needless to say, for most LA applications, you'll be sticking with `1`, as we do for the rest of the tutorial. A few example should make everything crystal clear:

![](http://mathurl.com/ldrpp8j.png)

```swift
var v = [1.0, 2.0]
var s = 3.0
var vsresult = [Double](count : v.count, repeatedValue : 0.0)
vDSP_vsaddD(v, 1, &s, &vsresult, 1, vDSP_Length(v.count))
vsresult    // returns [4.0, 5.0]
```

![](http://mathurl.com/p6uk25z.png)

```swift
vDSP_vsmulD(v, 1, &s, &vsresult, 1, vDSP_Length(v.count))
vsresult    // returns [3.0, 6.0]
```

![](http://mathurl.com/om9qrak.png)

```swift
vDSP_vsdivD(v, 1, &s, &vsresult, 1, vDSP_Length(v.count))
vsresult    // returns [0.333333333333333, 0.666666666666667]
```

Vector & Vector
--------------

Vector-vector operations pose no challenge to `Accelerate` and the associated functions look like

vDSP_v***D(vector_1, 1, vector_2, 1, &result, 1, length_of_vector)

Here are a few worked-out examples:

![](http://mathurl.com/kftp8ub.png)

```swift
var v1 = [2.0, 5.0]
var v2 = [3.0, 4.0]
var vvresult = [Double](count : 2, repeatedValue : 0.0)
vDSP_vaddD(v1, 1, v2, 1, &vvresult, 1, vDSP_Length(v1.count))
vvresult    // returns [5.0, 9.0]
```

![](http://mathurl.com/mkuokxm.png)
    
```swift
vDSP_vmulD(v1, 1, v2, 1, &vvresult, 1, vDSP_Length(v1.count))
vvresult    // returns [6.0, 20.0]
```

![](http://mathurl.com/mgqj5zx.png)

```swift
vDSP_vdivD(v1, 1, v2, 1, &vvresult, 1, vDSP_Length(v1.count))
vvresult    // returns [1.5, 0.8]
```

Dot Product
----------

![](http://mathurl.com/m8qlfvf.png)

```swift
var v3 = [1.0, 2.0]
var v4 = [3.0, 4.0]
var dpresult = 0.0
vDSP_dotprD(v3, 1, v4, 1, &dpresult, vDSP_Length(v3.count))
dpresult    // returns 11.0
```

Matrix Multiplication
-----------------

Matrices are passed into `Accelerate` as 1D arrays. As a result, matrix addition/subtraction is the same as vector addition/subtraction.

Matrix multiplication, on the other hand, is a bit more involved and requires this function:

```swift
vDSP_mmulD(matrix_1, 1, matrix_2, 1, &result, 1, 
           rows_of_matrix_1, columns_of_matrix_2, 
           columns_of_matrix_1_or_rows_of_matrix_2)
```

For example,

![](http://mathurl.com/mkx7vm4.png)
    
```swift
    var m1 = [ 3.0, 2.0, 4.0, 5.0, 6.0, 7.0 ]
    var m2 = [ 10.0, 20.0, 30.0, 30.0, 40.0, 50.0]
    var mresult = [Double](count : 9, repeatedValue : 0.0)

vDSP_mmulD(m1, 1, m2, 1, &mresult, 1, 3, 3, 2)
    mresult    // returns [90.0, 140.0, 190.0, 280.0, 370.0, 270.0, 400.0, 530.0]
```

Matrix Transpose
--------------

Matrix transpose can be obtained with

```swift
    vDSP_mtransD(matrix, 1, &result, 1, number_of_rows_of_result, number_of_columns_of_result)  
```

Like this,

![](http://mathurl.com/l65mgv5.png)

```swift
    var t = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
    var mtresult = [Double](count : t.count, repeatedValue : 0.0)
    vDSP_mtransD(t, 1, &mtresult, 1, 3, 2)
    mtresult    // returns [1.0, 4.0, 2.0, 5.0, 3.0, 6.0]
```

Matrix Inversion
---------------

Matrix inversion takes a bit more effort, but can be accomplished with the function below (see [Stack Overflow](http://stackoverflow.com/questions/11282746/how-to-perform-matrix-inverse-operation-using-the-accelerate-framework)):

![](http://mathurl.com/qgypepv.png)

```swift
    func invert(matrix : [Double]) -> [Double] {
        
        var inMatrix = matrix
        
        var pivot : __CLPK_integer = 0
        var workspace = 0.0
        var error : __CLPK_integer = 0
        
        var N = __CLPK_integer(sqrt(Double(matrix.count)))
        dgetrf_(&N, &N, &inMatrix, &N, &pivot, &error)
        
        if error != 0 {
            return inMatrix
        }
        
        dgetri_(&N, &inMatrix, &N, &pivot, &workspace, &N, &error)
        return inMatrix
    }

var m = [1.0, 2.0, 3.0, 4.0]
    invert(m)    // returns [-2.0, 1.0, 1.5, -0.5]
```

SwiftAccelerate

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